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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<a href="../../chapter_computational_complexity/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 2 章 複雜度分析
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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第 2 章 複雜度分析
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<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
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<span class="md-ellipsis">
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2.1 演算法效率評估
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</span>
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<span class="md-ellipsis">
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2.2 迭代與遞迴
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</span>
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<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.3 時間複雜度
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
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<span class="md-ellipsis">
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2.4 空間複雜度
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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2.5 小結
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</a>
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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第 3 章 資料結構
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</span>
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</a>
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<label class="md-nav__link " for="__nav_5" id="__nav_5_label" tabindex="0">
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_5">
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<span class="md-nav__icon md-icon"></span>
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第 3 章 資料結構
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
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<span class="md-ellipsis">
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3.1 資料結構分類
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<a href="../../chapter_data_structure/basic_data_types/" class="md-nav__link">
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<span class="md-ellipsis">
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3.2 基本資料型別
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<a href="../../chapter_data_structure/number_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.3 數字編碼 *
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<a href="../../chapter_data_structure/character_encoding/" class="md-nav__link">
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<span class="md-ellipsis">
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3.4 字元編碼 *
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<a href="../../chapter_data_structure/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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3.5 小結
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</span>
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</a>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<a href="../../chapter_array_and_linkedlist/" class="md-nav__link ">
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<span class="md-ellipsis">
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第 4 章 陣列與鏈結串列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_6" id="__nav_6_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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第 4 章 陣列與鏈結串列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
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<span class="md-ellipsis">
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4.1 陣列
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</span>
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.2 鏈結串列
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
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<span class="md-ellipsis">
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4.3 串列
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
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<span class="md-ellipsis">
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4.4 記憶體與快取 *
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</span>
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<span class="md-status md-status--new" title="最近新增">
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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4.5 小結
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</span>
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</a>
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</li>
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</ul>
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<a href="../../chapter_stack_and_queue/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38h12.56M6.77 14.32l.37-1.76 8.79 1.85-.37 1.76-8.79-1.85m1.16-4.21.76-1.61 8.14 3.78-.76 1.62-8.14-3.79m2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37-6.9-5.75m4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21 1.44-1.07M6.59 18.41v-1.8h8.98v1.8H6.59Z"/></svg>
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<span class="md-ellipsis">
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第 5 章 堆疊與佇列
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</span>
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</a>
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<label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
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第 5 章 堆疊與佇列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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<span class="md-ellipsis">
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5.1 堆疊
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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<span class="md-ellipsis">
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5.2 佇列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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<span class="md-ellipsis">
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5.3 雙向佇列
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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5.4 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_hashing/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21l-3.09-3.11m-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54h.03M19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2Z"/></svg>
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<span class="md-ellipsis">
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第 6 章 雜湊表
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</span>
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</a>
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<label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_8">
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<span class="md-nav__icon md-icon"></span>
|
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第 6 章 雜湊表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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<span class="md-ellipsis">
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6.1 雜湊表
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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<span class="md-ellipsis">
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6.2 雜湊衝突
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.3 雜湊演算法
|
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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<span class="md-ellipsis">
|
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6.4 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_tree/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1Z"/></svg>
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<span class="md-ellipsis">
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第 7 章 樹
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</span>
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</a>
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<label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_9">
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<span class="md-nav__icon md-icon"></span>
|
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|
第 7 章 樹
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.1 二元樹
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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7.2 二元樹走訪
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.3 二元樹陣列表示
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.4 二元搜尋樹
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
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<span class="md-ellipsis">
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7.5 AVL *
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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7.6 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_heap/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1Z"/></svg>
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<span class="md-ellipsis">
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第 8 章 堆積
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</span>
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</a>
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<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_10">
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<span class="md-nav__icon md-icon"></span>
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第 8 章 堆積
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_heap/heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.1 堆積
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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<span class="md-ellipsis">
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8.2 建堆積操作
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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<span class="md-ellipsis">
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8.3 Top-k 問題
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_heap/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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8.4 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_graph/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z"/></svg>
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<span class="md-ellipsis">
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第 9 章 圖
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</span>
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</a>
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<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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第 9 章 圖
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph/" class="md-nav__link">
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<span class="md-ellipsis">
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9.1 圖
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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9.2 圖基礎操作
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
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<span class="md-ellipsis">
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9.3 圖的走訪
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</span>
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</a>
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<li class="md-nav__item">
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<a href="../../chapter_graph/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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9.4 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_searching/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z"/></svg>
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<span class="md-ellipsis">
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第 10 章 搜尋
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</span>
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</a>
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<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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第 10 章 搜尋
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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<span class="md-ellipsis">
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10.1 二分搜尋
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_insertion/" class="md-nav__link">
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<span class="md-ellipsis">
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10.2 二分搜尋插入點
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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<span class="md-ellipsis">
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10.3 二分搜尋邊界
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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<span class="md-ellipsis">
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10.4 雜湊最佳化策略
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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<span class="md-ellipsis">
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10.5 重識搜尋演算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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10.6 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
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<div class="md-nav__link md-nav__container">
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<a href="../" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2v-2Z"/></svg>
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<span class="md-ellipsis">
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第 11 章 排序
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</span>
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</a>
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<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_13">
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<span class="md-nav__icon md-icon"></span>
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第 11 章 排序
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../sorting_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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11.1 排序演算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../selection_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.2 選擇排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../bubble_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.3 泡沫排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../insertion_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.4 插入排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../quick_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.5 快速排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../merge_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.6 合併排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../heap_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.7 堆積排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../bucket_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.8 桶排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../counting_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.9 計數排序
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../radix_sort/" class="md-nav__link">
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<span class="md-ellipsis">
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11.10 基數排序
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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<span class="md-ellipsis">
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11.11 小結
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<a href="./" class="md-nav__link md-nav__link--active">
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<span class="md-ellipsis">
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11.11 小結
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目錄">
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<label class="md-nav__title" for="__toc">
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<span class="md-nav__icon md-icon"></span>
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目錄
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</label>
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="#1" class="md-nav__link">
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<span class="md-ellipsis">
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|
1. 重點回顧
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="#2-q-a" class="md-nav__link">
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<span class="md-ellipsis">
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2. Q & A
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_divide_and_conquer/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z"/></svg>
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<span class="md-ellipsis">
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第 12 章 分治
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</span>
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</a>
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<label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_14">
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<span class="md-nav__icon md-icon"></span>
|
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第 12 章 分治
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
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<span class="md-ellipsis">
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12.1 分治演算法
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
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<span class="md-ellipsis">
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12.2 分治搜尋策略
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.3 構建樹問題
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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12.4 河內塔問題
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_divide_and_conquer/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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12.5 小結
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
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<div class="md-nav__link md-nav__container">
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<a href="../../chapter_backtracking/" class="md-nav__link ">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z"/></svg>
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<span class="md-ellipsis">
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|
第 13 章 回溯
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|
|
</span>
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|
</a>
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<label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
|
|
|
<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_15">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
第 13 章 回溯
|
|
|
</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
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<span class="md-ellipsis">
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13.1 回溯演算法
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<span class="md-ellipsis">
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13.2 全排列問題
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.3 子集和問題
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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13.4 N 皇后問題
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<span class="md-ellipsis">
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13.5 小結
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</a>
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<span class="md-ellipsis">
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第 14 章 動態規劃
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<span class="md-nav__icon md-icon"></span>
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第 14 章 動態規劃
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<span class="md-ellipsis">
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14.1 初探動態規劃
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<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
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<span class="md-ellipsis">
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14.2 DP 問題特性
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<a href="../../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
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<span class="md-ellipsis">
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14.3 DP 解題思路
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<span class="md-ellipsis">
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14.4 0-1 背包問題
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<a href="../../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.5 完全背包問題
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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14.6 編輯距離問題
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</span>
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<a href="../../chapter_dynamic_programming/summary/" class="md-nav__link">
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<span class="md-ellipsis">
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14.7 小結
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</a>
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<span class="md-ellipsis">
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第 15 章 貪婪
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</span>
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</a>
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<span class="md-nav__icon md-icon"></span>
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第 15 章 貪婪
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<span class="md-ellipsis">
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15.1 貪婪演算法
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</span>
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<a href="../../chapter_greedy/fractional_knapsack_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.2 分數背包問題
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</span>
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<a href="../../chapter_greedy/max_capacity_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.3 最大容量問題
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</span>
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<li class="md-nav__item">
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<a href="../../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">
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<span class="md-ellipsis">
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15.4 最大切分乘積問題
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</span>
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<li class="md-nav__item">
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<span class="md-ellipsis">
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15.5 小結
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</span>
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</a>
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<span class="md-ellipsis">
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第 16 章 附錄
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</span>
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</a>
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第 16 章 附錄
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16.1 程式設計環境安裝
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<h1 id="1111">11.11 小結<a class="headerlink" href="#1111" title="Permanent link">¶</a></h1>
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<h3 id="1">1. 重點回顧<a class="headerlink" href="#1" title="Permanent link">¶</a></h3>
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<ul>
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<li>泡沫排序透過交換相鄰元素來實現排序。透過新增一個標誌位來實現提前返回，我們可以將泡沫排序的最佳時間複雜度最佳化到 <span class="arithmatex">\(O(n)\)</span> 。</li>
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<li>插入排序每輪將未排序區間內的元素插入到已排序區間的正確位置，從而完成排序。雖然插入排序的時間複雜度為 <span class="arithmatex">\(O(n^2)\)</span> ，但由於單元操作相對較少，因此在小資料量的排序任務中非常受歡迎。</li>
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<li>快速排序基於哨兵劃分操作實現排序。在哨兵劃分中，有可能每次都選取到最差的基準數，導致時間複雜度劣化至 <span class="arithmatex">\(O(n^2)\)</span> 。引入中位數基準數或隨機基準數可以降低這種劣化的機率。尾遞迴方法可以有效地減少遞迴深度，將空間複雜度最佳化到 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
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<li>合併排序包括劃分和合並兩個階段，典型地體現了分治策略。在合併排序中，排序陣列需要建立輔助陣列，空間複雜度為 <span class="arithmatex">\(O(n)\)</span> ；然而排序鏈結串列的空間複雜度可以最佳化至 <span class="arithmatex">\(O(1)\)</span> 。</li>
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<li>桶排序包含三個步驟：資料分桶、桶內排序和合並結果。它同樣體現了分治策略，適用於資料體量很大的情況。桶排序的關鍵在於對資料進行平均分配。</li>
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<li>計數排序是桶排序的一個特例，它透過統計資料出現的次數來實現排序。計數排序適用於資料量大但資料範圍有限的情況，並且要求資料能夠轉換為正整數。</li>
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<li>基數排序透過逐位排序來實現資料排序，要求資料能夠表示為固定位數的數字。</li>
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<li>總的來說，我們希望找到一種排序演算法，具有高效率、穩定、原地以及正向自適應性等優點。然而，正如其他資料結構和演算法一樣，沒有一種排序演算法能夠同時滿足所有這些條件。在實際應用中，我們需要根據資料的特性來選擇合適的排序演算法。</li>
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<li>圖 11-19 對比了主流排序演算法的效率、穩定性、就地性和自適應性等。</li>
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</ul>
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<p><a class="glightbox" href="../summary.assets/sorting_algorithms_comparison.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="排序演算法對比" class="animation-figure" src="../summary.assets/sorting_algorithms_comparison.png" /></a></p>
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<p align="center"> 圖 11-19 排序演算法對比 </p>
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<h3 id="2-q-a">2. Q & A<a class="headerlink" href="#2-q-a" title="Permanent link">¶</a></h3>
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<p><strong>Q</strong>：排序演算法穩定性在什麼情況下是必需的？</p>
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<p>在現實中，我們有可能基於物件的某個屬性進行排序。例如，學生有姓名和身高兩個屬性，我們希望實現一個多級排序：先按照姓名進行排序，得到 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> ；再對身高進行排序。由於排序演算法不穩定，因此可能得到 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> 。</p>
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<p>可以發現，學生 D 和 C 的位置發生了交換，姓名的有序性被破壞了，而這是我們不希望看到的。</p>
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<p><strong>Q</strong>：哨兵劃分中“從右往左查詢”與“從左往右查詢”的順序可以交換嗎？</p>
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<p>不行，當我們以最左端元素為基準數時，必須先“從右往左查詢”再“從左往右查詢”。這個結論有些反直覺，我們來剖析一下原因。</p>
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<p>哨兵劃分 <code>partition()</code> 的最後一步是交換 <code>nums[left]</code> 和 <code>nums[i]</code> 。完成交換後，基準數左邊的元素都 <code><=</code> 基準數，<strong>這就要求最後一步交換前 <code>nums[left] >= nums[i]</code> 必須成立</strong>。假設我們先“從左往右查詢”，那麼如果找不到比基準數更大的元素，<strong>則會在 <code>i == j</code> 時跳出迴圈，此時可能 <code>nums[j] == nums[i] > nums[left]</code></strong>。也就是說，此時最後一步交換操作會把一個比基準數更大的元素交換至陣列最左端，導致哨兵劃分失敗。</p>
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<p>舉個例子，給定陣列 <code>[0, 0, 0, 0, 1]</code> ，如果先“從左向右查詢”，哨兵劃分後陣列為 <code>[1, 0, 0, 0, 0]</code> ，這個結果是不正確的。</p>
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<p>再深入思考一下，如果我們選擇 <code>nums[right]</code> 為基準數，那麼正好反過來，必須先“從左往右查詢”。</p>
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<p><strong>Q</strong>：關於尾遞迴最佳化，為什麼選短的陣列能保證遞迴深度不超過 <span class="arithmatex">\(\log n\)</span> ？</p>
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<p>遞迴深度就是當前未返回的遞迴方法的數量。每輪哨兵劃分我們將原陣列劃分為兩個子陣列。在尾遞迴最佳化後，向下遞迴的子陣列長度最大為原陣列長度的一半。假設最差情況，一直為一半長度，那麼最終的遞迴深度就是 <span class="arithmatex">\(\log n\)</span> 。</p>
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<p>回顧原始的快速排序，我們有可能會連續地遞迴長度較大的陣列，最差情況下為 <span class="arithmatex">\(n\)</span>、<span class="arithmatex">\(n - 1\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</span> ，遞迴深度為 <span class="arithmatex">\(n\)</span> 。尾遞迴最佳化可以避免這種情況出現。</p>
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<p><strong>Q</strong>：當陣列中所有元素都相等時，快速排序的時間複雜度是 <span class="arithmatex">\(O(n^2)\)</span> 嗎？該如何處理這種退化情況？</p>
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<p>是的。對於這種情況，可以考慮透過哨兵劃分將陣列劃分為三個部分：小於、等於、大於基準數。僅向下遞迴小於和大於的兩部分。在該方法下，輸入元素全部相等的陣列，僅一輪哨兵劃分即可完成排序。</p>
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<p><strong>Q</strong>：桶排序的最差時間複雜度為什麼是 <span class="arithmatex">\(O(n^2)\)</span> ？</p>
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<p>最差情況下，所有元素被分至同一個桶中。如果我們採用一個 <span class="arithmatex">\(O(n^2)\)</span> 演算法來排序這些元素，則時間複雜度為 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
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